Extrinsic geometry of calibrated submanifolds

TL;DR

This paper introduces a Lie-theoretic condition called compliancy for calibrations, characterizes it for many geometric cases, and explores how it influences the extrinsic geometry of calibrated submanifolds.

Contribution

It defines and characterizes the compliancy condition for calibrations and relates it to the extrinsic geometry of calibrated immersions, generalizing superminimal surfaces.

Findings

Compliancy holds for Kähler, special Lagrangian, associative, coassociative, and Cayley calibrations.

Provides a sufficient condition for compliancy based on natural involutions.

Characterizes extrinsic geometry conditions for calibrated immersions with parallel, compliant calibrations.

Abstract

Given a calibration $\alpha$ whose stabilizer acts transitively on the Grassmanian of calibrated planes, we introduce a nontrivial Lie-theoretic condition on $\alpha$, which we call compliancy, and show that this condition holds for many interesting geometric calibrations, including K\"ahler, special Lagrangian, associative, coassociative, and Cayley. We determine a sufficient condition that ensures compliancy of $\alpha$, we completely characterize compliancy in terms of properties of a natural involution determined by a calibrated plane, and we relate compliancy to the geometry of the calibrated Grassmanian. The condition that a Riemannian immersion $\iota \colon L \to M$ be calibrated is a first order condition. By contrast, its extrinsic geometry, given by the second fundamental form $A$ and the induced tangent and normal connections $\nabla$ on $TL$ and $D$ on $NL$, respectively, is second order information. We characterize the conditions imposed on the extrinsic geometric data $(A, \nabla, D)$ when the Riemannian immersion $\iota \colon L \to M$ is calibrated with respect to a calibration $\alpha$ on $M$ which is both parallel and compliant. This motivate the definition of an infinitesimally calibrated Riemannian immersion, generalizing the classical notion of a superminimal surface in $\mathbb R^4$.